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In [[mathematics]], the '''Pontryagin classes''' are certain [[characteristic class]]es. The Pontryagin class lies in [[cohomology group]]s with degree a multiple of four. It applies to real [[vector bundle]]s.
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== Definition ==
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Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class ''p<sub>k</sub>''(''E'') is defined as
:''p<sub>k</sub>''(''E'') = ''p<sub>k</sub>''(''E'', '''Z''') = (−1)<sup>''k''</sup> ''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') ∈ ''H''<sup>4''k''</sup>(''M'', '''Z'''),
where:
*''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') denotes the 2''k''-th [[Chern class]] of the [[complexification]] ''E'' ⊗ '''C''' = ''E'' ⊕ ''iE'' of ''E'',
*''H''<sup>4''k''</sup>(''M'', '''Z''') is the 4''k''-[[cohomology]] group of ''M'' with [[integer]] coefficients.
 
The rational Pontryagin class ''p<sub>k</sub>''(''E'', '''Q''') is defined to be the image of ''p<sub>k</sub>''(''E'') in ''H''<sup>4''k''</sup>(''M'', '''Q'''), the 4''k''-[[cohomology]] group of ''M'' with [[Rational number|rational]] coefficients.
 
== Properties ==
The '''total Pontryagin class'''
:<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\mathbf{Z}),</math>
is (modulo 2-torsion) multiplicative with respect to  
[[Glossary of differential geometry and topology#W|Whitney sum]] of vector bundles, i.e.,
:<math>2p(E\oplus F)=2p(E)\smile p(F)</math>
for two vector bundles ''E'' and ''F'' over ''M''.  In terms of the individual Pontryagin classes ''p<sub>k</sub>'',
:<math>2p_1(E\oplus F)=2p_1(E)+2p_1(F),</math>
:<math>2p_2(E\oplus F)=2p_2(E)+2p_1(E)\smile p_1(F)+2p_2(F)</math>
and so on.  
 
The vanishing of the Pontryagin classes and [[Stiefel-Whitney class]]es of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to [[Vector_bundle#Vector_bundle_morphisms|vector bundle isomorphism]], there is a unique nontrivial rank 10 vector bundle ''E''<sub>10</sub> over the [[N-sphere|9-sphere]]. (The [[clutching function]] for ''E''<sub>10</sub> arises from the [[Orthogonal_group#Homotopy_groups|stable homotopy group]] π<sub>8</sub>(O(10)) = '''Z'''/2'''Z'''.)  The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the [[Stiefel-Whitney class]] ''w''<sub>9</sub> of ''E''<sub>10</sub> vanishes by the [[Stiefel-Whitney class#Relations_over_the_Steenrod_algebra|Wu formula]] ''w''<sub>9</sub> = ''w''<sub>1</sub>''w''<sub>8</sub> + Sq<sup>1</sup>(''w''<sub>8</sub>).  Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of ''E''<sub>10</sub> with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}
 
Given a 2''k''-dimensional vector bundle ''E'' we have  
:<math>p_k(E)=e(E)\smile e(E),</math>
where ''e''(''E'') denotes the [[Euler class]] of ''E'', and <math>\smile</math> denotes the [[cup product]] of cohomology classes; under the [[splitting principle]], this corresponds to the square of the [[Vandermonde polynomial]] being the [[discriminant]].
 
=== Pontryagin classes and curvature ===
As was shown by [[Shiing-Shen Chern]] and [[André Weil]] around 1948, the rational Pontryagin classes
:<math>p_k(E,\mathbf{Q})\in H^{4k}(M,\mathbf{Q})</math>
can be presented as differential forms which depend polynomially on the [[curvature form]] of a vector bundle. This [[Chern-Weil theory]] revealed a major connection between algebraic topology and global differential geometry.
 
For a [[vector bundle]] ''E'' over a ''n''-dimensional [[differentiable manifold]] ''M'' equipped with a [[connection form|connection]], the total Pontryagin class is expressed as
:<math>p=\left[1-\frac{{\rm Tr}(\Omega ^2)}{8 \pi ^2}+\frac{{\rm Tr}(\Omega ^2)^2-2 {\rm Tr}(\Omega ^4)}{128 \pi ^4}-\frac{{\rm Tr}(\Omega ^2)^3-6 {\rm Tr}(\Omega ^2) {\rm Tr}(\Omega ^4)+8 {\rm Tr}(\Omega ^6)}{3072 \pi ^6}+\cdots\right]\in H^*_{dR}(M),</math>
 
where Ω denotes the [[curvature form]], and ''H*''<sub>dR</sub>(''M'') denotes the [[de Rham cohomology]] groups.{{Citation needed|date=July 2009}}
 
=== Pontryagin classes of a manifold ===
The '''Pontryagin classes of a smooth manifold''' are defined to be the Pontryagin classes of its [[tangent bundle]].
 
[[Sergei Novikov (mathematician)|Novikov]] proved in 1966 that if manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes ''p<sub>k</sub>''(''M'', '''Q''') in ''H''<sup>4''k''</sup>(''M'', '''Q''') are the same.
 
If the dimension is at least five, there are at most finitely many different smooth manifolds with given [[Homotopy#Homotopy equivalence of spaces|homotopy type]] and Pontryagin classes.
 
== Pontryagin numbers ==
'''Pontryagin numbers''' are certain [[topological invariant]]s of a smooth [[manifold]]. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a [[manifold]] as follows:
 
Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers
:''k''<sub>1</sub>, ''k''<sub>2</sub>, ..., ''k<sub>m</sub>'' such that ''k''<sub>1</sub>+''k''<sub>2</sub>+...+''k<sub>m</sub>'' =''n''.
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
where ''p<sub>k</sub>'' denotes the ''k''-th Pontryagin class and [''M''] the [[fundamental class]] of ''M''.
 
=== Properties ===
#Pontryagin numbers are oriented [[cobordism]] invariant; and together with [[Stiefel-Whitney number]]s they determine an oriented manifold's oriented cobordism class.
#Pontryagin numbers of closed Riemannian manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
#Such invariants as [[Signature (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers.
 
== Generalizations ==
There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure.
 
== See also ==
*[[Chern-Simons form]]
 
== References ==
*{{cite book
  |author= [[John Milnor|Milnor John W.]]
  |coauthors= [[Jim Stasheff|Stasheff, James D.]]
  |title= Characteristic classes
  |publisher= Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo
  |year= 1974
  |isbn=  0-691-08122-0}}
* {{Cite journal | last=Hatcher | first=Allen | author-link=Allen Hatcher  | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | ref=harv | postscript=<!--None--> | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}}
 
==External links==
* {{springer|title=Pontryagin class|id=p/p073750}}
 
[[Category:Characteristic classes]]
[[Category:Differential topology]]

Latest revision as of 20:35, 1 December 2014

For your own offense, you might eat Gunboats which can easily shoot at enemy defenses coming from a range and Landing Special creations which you must load when you train types for example Rifleman, Heavy, Zooka, Warrior and Takes a dive. To your village defenses, you might carry structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Catapult Launcher to assist your entire family eradicate enemies.

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Gemstone are known as typically the games primary forex. The Jewels are used to purchase resources along among speeding up numerous powerful tasks. The Treasures can also be which is used to buy bonus items. Apart from that, it can possibly let the leader noticeable any undesired debris as a way to obtain a a lot more gems. Players can quickly obtain Gems through undertaking numerous tasks or certainly using the clash of clans get into available online.

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